How accurate should one expect autofocus to be?

This is my first post here. I have just recently bought a used D800 (with a AF-S Nikkor 24-70mm 1:2.8G). I intend to use it mainly for taking photos of my family and maybe some nature photos if I ever get the time off. I am a beginner photographer, and this is my first real experience with an autofocus SLR (having previously mainly used a manual focus Minolta film SLR).

Now to my question: I'm having problems getting sharp photos. One typical example would be taking photos of my 2 year old. I set the camera at 70mm, 1/250 (she's never still), ƒ/2.8, AF-C, d9 area mode and keep the AF-point centered on her eye while keeping AF-ON depressed (I have disabled shutter half-press AF). I then wait a few seconds for the right moment and release the shutter. The result is a perfectly sharp eye (when viewed at 100 %) only in about 10 % of the cases. The rest are either significantly back- or front focused (seemingly at random).

Having never operated such a camera before, I don't know if there is anything wrong with my settings, my technique, my equipment or if the results are just what one should expect. All advice is highly appreciated.